This paper proposes an over-the-air aggregation framework for federated learning (FL) in broadband wireless networks where not only edge devices but also a base station (BS) has its own local dataset. The proposed framework leverages the BS dataset to improve communication efficiency of FL by reducing the number of channel uses required for the model convergence as well as avoiding the signaling overhead incurred by power scale coordination among edge devices. We analyze the convergence to a stationary point without convexity assumption on the objective function. The analysis result reveals that the utilization of BS dataset improves the convergence rate and the update distortion caused by the limited power budget is a crucial factor hindering the model convergence. To facilitate the convergence, we develop an optimized power control method by solving the distortion minimization problem without assumptions on power scale coordination and global CSI at BS. Our simulation results validate that BS dataset is beneficial to reducing the number of channel uses for the model convergence and the developed power control method outperforms the conventional method in terms of both convergence rate and converged test accuracy. Furthermore, we identify some scenarios where the compression of local update can be helpful to reduce communication resources for model training.